How is the back-button focus any different than pressing the shutter button halfway on subject, composing your shot with, then depressing fully??
Ok here goes
Situation 1 - shooting in a studio narrow aperture your distance to the subject not changing much you can focus onece then just shoot without having to keep refocusing if you are not moving, if you change position or distance you can focus again then shoot off a few shots.
Situation 2 - Shooting in AI servo you just hold that button down the whole time to make sure it holds tracking and press the trigger when you want (its much much better than the half pressing).
Situation 3 - Landscape on a tripod, frame the shot focus using the back button recompose to suit and now you can shoot at will without it trying to refocus on something
Situation 4 - use AF to aquire the target then MF to refine in this case hitting the shutter button doesnt try to refocus.
There are probably more but those are a few handy situations.